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Linearization of bounded holomorphic mappings on Banach spaces. (English) Zbl 0747.46038
The author shows that for every open subset $$U$$ of a Banach space there exists a unique Banach space $$G^ \infty(U)$$ and a holomorphic mapping $$g_ u:U\to G^ \infty(U)$$ such that every Banach valued bounded holomorphic function on $$U$$ can be written as a composition of $$g_ u$$ and a Banach valued continuous linear mapping on $$G^ \infty(U)$$. This gives a linearization of bounded holomorphic mappings and shows that $$H^ \infty(U)$$ has the structure of a dual Banach space. Applications to the study of holomorphic mappings of compact type, the approximation property and polynomials are given using this linearization result.
Reviewer: S.Dineen

##### MSC:
 46G20 Infinite-dimensional holomorphy 46E50 Spaces of differentiable or holomorphic functions on infinite-dimensional spaces 46B28 Spaces of operators; tensor products; approximation properties
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